The present invention relates to data transmission methods of transmitting data in blocks, and more particularly to a data transmission method in which both random error and burst error in a transmission system can be corrected. The term "transmission" as herein used is intended to include recording and reproducing as well as direct transmission through a communications path.
In a conventional data transmission method of this general type, data is arranged two dimensionally in matrix form, and check words are applied in horizontal (x) and vertical (y) directions in such a manner that an error detecting code is formed in the horizontal .(x) direction and an error correcting code is formed in the vertical (y) direction before the data is transmitted. A typical data arrangement used with a stationary head multi-track recorder is shown in FIG. 1 as an example of such a method. In a stationary head multi-track recorder, the pattern shown in FIG. 1 is recorded on the tape. In the case where it is applied to a single track or single channel transmission system, the words W.sub.11, W.sub.12 . . . W.sub.1l, CRC.sub.l, W.sub.21, W.sub.22, . . . W.sub.2l, CRC.sub.2, --- , W.sub.m1, W.sub.m2, W.sub.ml, CRC.sub.m, P.sub.1, P.sub.2 . . . P.sub.l, CRC.sub.m+1, Q.sub.1, Q.sub.2, . . . Q.sub.l, and CRC.sub.m+2 are transmitted word by word in the stated order. In FIG. 1, the data of l.times.m words is arranged two dimensionally in a matrix form, words horizontally and m words vertically. The words CRC.sub.1 through CRC.sub.m+2 are check words for detecting errors in the horizontal lines, and are generally 16 bits each. The other words (other than CRC) are k bits each. Furthermore, the words P.sub.x and Q.sub.x (where x=1 through l) are words for correcting errors in the vertical columns, forming Reed-Solomon codes in the vertical columns. The Reed-Solomon codes P.sub.x and Q.sub.x are so formed as to meet the following expressions: ##EQU1## where addition should be performed with modulo 2, and .alpha. is a Galois field primative element GF(2.sup.k) which during the tranmission is corrected as follows: On the data receiving side, with respect to error-including data and a check word, syndromes S.sub.Px and S.sub.Qx as defined below are obtained for each column: ##EQU2##
It is assumed that, in the x-th column, two errors occur in the i-th and j-th words (i, j=1 to m), as a result of which W.sub.ix and W.sub.ij are changed to W.sub.ix '= W.sub.ix +e.sub.ix and W.sub.jx '=W.sub.jx +e.sub.jx (where e.sub.ix and e.sub.jx are the error patterns of W.sub.ix and W.sub.jx, respectively).
Therefore: EQU S.sub.Px =e.sub.ix +e.sub.jx EQU S.sub.Qx =.alpha..sup.m+2=i e.sub.ix +a.sup.m+2- e.sub.jx.
From these expressions, ##EQU3## The errors of the lines are detected with CRC.sub.y (where y=1 to m+2), and the error positions i and j obtained. Therefore, the error patterns e.sub.ix and e.sub.jx can be obtained from the above expressions. The data values e.sub.ix and e.sub.jx thus obtained are added to the erroneous data W.sub.ix ' and W.sub.jx ' to obtain correct data as follows: EQU W.sub.ix '+e.sub.ix =W.sub.ix +e.sub.ix +e.sub.ix =W.sub.ix EQU W.sub.jx '+e.sub.jx =W.sub.jx +e.sub.jx +e.sub.jx =W.sub.jx.
In the case where the number of erroneous words is only one, one of the data values e.sub.ix and e.sub.jx should be considered as zero.
In the conventional data transmission method employing the data arrangement as shown in FIG. 1, if the number of lines in which errors occur is two or less, the errors can be corrected, even if they are burst errors of long lengths. However, in the case where the number of lines including errors is more than two, with the described method it is impossible to detect the columns in which the errors occurred, and therefore it is impossible to correct an error pattern as shown in FIGS. 2A and 2B in which the number of errors in a single column is two or less. In FIGS. 2A and 2B, the mark x indicates the positions of erroneous data, and the numeral "1" indicates positive results of error detection by CRC, namely, that error exists.
Accordingly, in the case of a transmission system in which random errors of short length occur frequently, the number of uncorrectable errors is increased and thus it is impossible to correctly transmit data. In the case where the data has a high degree of correlation as in the case of PCM audio data, the effect of uncorrectable errors can be somewhat decreased by a correcting method such as an average interpolation method. However, in the case of data such as computer software, the erroneous data cannot be corrected in this manner, thus resulting in the erroneous operation of the system. Accordingly, the data transmission method employing the data arrangement as shown in FIG. 1 is unsuitable for the transmission of computer software in a transmission system in which random errors occur frequently.